All group‐based latin squares possess near transversals

Goddyn, Luis; Halasz, Kevin

doi:10.1002/jcd.21701

Cited by 4 publications

(5 citation statements)

References 22 publications

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“…Using Lemma 3, we have that in this case all d-iterated quasigroups have near transversals. For groups, the Brualdi's conjecture is true by the result of [8].…”

Section: Proofs Of the Main Resultsmentioning

confidence: 96%

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Transversals, near transversals, and diagonals in iterated groups and quasigroups

Taranenko¹

2020

Preprint

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Given a binary quasigroup G of order n, a d-iterated quasigroup G [d] is the (d + 1)-ary quasigroup equal to the d-times composition of G with itself. The Cayley table of every d-ary quasigroup is a d-dimensional latin hypercube. Transversals and diagonals in multiary quasigroups are defined so that to coincide with those in the corresponding latin hypercube.We prove that if a group G of order n satisfies the Hall-Paige condition, then the number of transversals in 1)) for large d, where G ′ is the commutator subgroup of G. For a general quasigroup G, we obtain similar estimations on the numbers of transversals and near transversals in G[d] and develop a method for counting diagonals of other types in iterated quasigroups.

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“…Using Lemma 3, we have that in this case all d-iterated quasigroups have near transversals. For groups, the Brualdi's conjecture is true by the result of [8].…”

Section: Proofs Of the Main Resultsmentioning

confidence: 96%

“…Concerning the Brualdi's conjecture, in preprint [8] it was proved that the Cayley table of every group contains a transversal.…”

Section: Introductionmentioning

confidence: 99%

Transversals, near transversals, and diagonals in iterated groups and quasigroups

Taranenko¹

2020

Preprint

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“…Using Lemma 19, we have that in this case all d-iterated quasigroups have near transversals. For groups, Brualdi's conjecture is true by [12].…”

Section: Discussionmentioning

confidence: 99%

“…Recently in preprint [8] there was given an alternative (asymptotic) proof for the conjecture. Concerning Brualdi's conjecture, in [12] it was proved that the Cayley table of every group contains a near transversal.…”

Section: Introductionmentioning

confidence: 99%

Transversals, Near Transversals, and Diagonals in Iterated Groups and Quasigroups

Taranenko

2021

Electron. J. Combin.

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Given a binary quasigroup $G$ of order $n$, a $d$-iterated quasigroup $G[d]$ is the $(d+1)$-ary quasigroup equal to the $d$-times composition of $G$ with itself. The Cayley table of every $d$-ary quasigroup is a $d$-dimensional latin hypercube. Transversals and diagonals in multiary quasigroups are defined so as to coincide with those in the corresponding latin hypercube. We prove that if a group $G$ of order $n$ satisfies the Hall–Paige condition, then the number of transversals in $G[d]$ is equal to $ \frac{n!}{ |G'| n^{n-1}} \cdot n!^{d} (1 + o(1))$ for large $d$, where $G'$ is the commutator subgroup of $G$. For a general quasigroup $G$, we obtain similar estimations on the numbers of transversals and near transversals in $G[d]$ and develop a method for counting diagonals of other types in iterated quasigroups.

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“…The following conjecture has been attributed to Brualdi (see [11, p. 103]) and Stein [25] and, in [13], to Ryser. It has recently been proved for Cayley tables of finite groups [17]. For several generalisations of the conjecture in terms of hypergraphs, see [1] (although some of those generalisations have since been shown to fail [16]).…”

Section: Introductionmentioning

confidence: 99%